Solve -4/3m^2+4m=-2/3m^2+2

$$- \frac { 4 } { 3 } m ^ { 2 } + 4 m = - \frac { 2 } { 3 } m ^ { 2 } + 2$$

Solve for m

$$m=\sqrt{6}+3\approx 5.449489743$$

$$m=3-\sqrt{6}\approx 0.550510257$$

Steps Using the Quadratic Formula

Add \frac{2}{3}m^{2} to both sides.

$$-\frac{4}{3}m^{2}+4m+\frac{2}{3}m^{2}=2$$

Combine -\frac{4}{3}m^{2} and \frac{2}{3}m^{2} to get -\frac{2}{3}m^{2}.

$$-\frac{2}{3}m^{2}+4m=2$$

Subtract 2 from both sides.

$$-\frac{2}{3}m^{2}+4m-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{2}{3} for a, 4 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-4±\sqrt{4^{2}-4\left(-\frac{2}{3}\right)\left(-2\right)}}{2\left(-\frac{2}{3}\right)}$$

Square 4.

$$m=\frac{-4±\sqrt{16-4\left(-\frac{2}{3}\right)\left(-2\right)}}{2\left(-\frac{2}{3}\right)}$$

Multiply -4 times -\frac{2}{3}.

$$m=\frac{-4±\sqrt{16+\frac{8}{3}\left(-2\right)}}{2\left(-\frac{2}{3}\right)}$$

Multiply \frac{8}{3} times -2.

$$m=\frac{-4±\sqrt{16-\frac{16}{3}}}{2\left(-\frac{2}{3}\right)}$$

Add 16 to -\frac{16}{3}.

$$m=\frac{-4±\sqrt{\frac{32}{3}}}{2\left(-\frac{2}{3}\right)}$$

Take the square root of \frac{32}{3}.

$$m=\frac{-4±\frac{4\sqrt{6}}{3}}{2\left(-\frac{2}{3}\right)}$$

Multiply 2 times -\frac{2}{3}.

$$m=\frac{-4±\frac{4\sqrt{6}}{3}}{-\frac{4}{3}}$$

Now solve the equation m=\frac{-4±\frac{4\sqrt{6}}{3}}{-\frac{4}{3}} when ± is plus. Add -4 to \frac{4\sqrt{6}}{3}.

$$m=\frac{\frac{4\sqrt{6}}{3}-4}{-\frac{4}{3}}$$

Divide -4+\frac{4\sqrt{6}}{3} by -\frac{4}{3} by multiplying -4+\frac{4\sqrt{6}}{3} by the reciprocal of -\frac{4}{3}.

$$m=3-\sqrt{6}$$

Now solve the equation m=\frac{-4±\frac{4\sqrt{6}}{3}}{-\frac{4}{3}} when ± is minus. Subtract \frac{4\sqrt{6}}{3} from -4.

$$m=\frac{-\frac{4\sqrt{6}}{3}-4}{-\frac{4}{3}}$$

Divide -4-\frac{4\sqrt{6}}{3} by -\frac{4}{3} by multiplying -4-\frac{4\sqrt{6}}{3} by the reciprocal of -\frac{4}{3}.

$$m=\sqrt{6}+3$$

The equation is now solved.

$$m=3-\sqrt{6} m=\sqrt{6}+3$$

Steps for Completing the Square

Add \frac{2}{3}m^{2} to both sides.

$$-\frac{4}{3}m^{2}+4m+\frac{2}{3}m^{2}=2$$

Combine -\frac{4}{3}m^{2} and \frac{2}{3}m^{2} to get -\frac{2}{3}m^{2}.

$$-\frac{2}{3}m^{2}+4m=2$$

Divide both sides of the equation by -\frac{2}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{2}{3}m^{2}+4m}{-\frac{2}{3}}=\frac{2}{-\frac{2}{3}}$$

Dividing by -\frac{2}{3} undoes the multiplication by -\frac{2}{3}.

$$m^{2}+\frac{4}{-\frac{2}{3}}m=\frac{2}{-\frac{2}{3}}$$

Divide 4 by -\frac{2}{3} by multiplying 4 by the reciprocal of -\frac{2}{3}.

$$m^{2}-6m=\frac{2}{-\frac{2}{3}}$$

Divide 2 by -\frac{2}{3} by multiplying 2 by the reciprocal of -\frac{2}{3}.

$$m^{2}-6m=-3$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-6m+\left(-3\right)^{2}=-3+\left(-3\right)^{2}$$

Square -3.

$$m^{2}-6m+9=-3+9$$

Add -3 to 9.

$$m^{2}-6m+9=6$$

Factor m^{2}-6m+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-3\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-3\right)^{2}}=\sqrt{6}$$

Simplify.

$$m-3=\sqrt{6} m-3=-\sqrt{6}$$

Add 3 to both sides of the equation.

$$m=\sqrt{6}+3 m=3-\sqrt{6}$$